A fully coupled thermal-electrochemical-structural–pore pressure analysis is used when
the mechanical, thermal, electrical, ion concentration, and fluid flow fields affect
each other strongly.
You must use coupled structural-thermal-electrochemical–pore pressure elements in this
type of analysis.
A fully coupled thermal-electrochemical-structural–pore pressure analysis is used for the
analysis of battery electrochemistry applications that require solving simultaneously
for the following:
displacements,
temperature,
electric potentials in the solid electrodes,
electric potential in the electrolyte,
concentration of ions in the electrolyte,
concentration in the solid particles used in the electrodes, and
fluid pore pressure mediated electrolyte flow with or without convective
effects. Particle swelling and element deformation can affect the electrolyte
convective flow and fluid pressure.
A complete understanding of battery electrochemistry requires modeling the mutual influence of
the deformation of the porous electrodes and the flow of the ion-carrying electrolyte
within the interconnected pores. This coupling is an important aspect of the battery
multiphysics, and you can use the fully coupled thermal-electrochemical-structural–pore
pressure procedure to better understand the influence of this coupling on the battery
performance. A fully coupled thermal-electrochemical-structural–pore pressure analysis
is a monolithic solve of the coupled pore fluid diffusion-displacement (see Coupled Pore Fluid Diffusion and Stress Analysis) and coupled
thermal-electrochemical-structural (see Fully Coupled Thermal-Electrochemical-Structural Analysis) analyses.
The primary example of a fully coupled thermal-electrochemical-structural–pore pressure
analysis in a battery electrochemistry application is the charging and discharging
of lithium ion battery cells. During the charging cycle, the lithium ions are
extracted (deintercalated) from the active particles of the positive electrode
(cathode). This process results in a reduction of the volume of the active
particles, which in turn leads to an increase in porosity as well as flow of
electrolyte into the cathode. The ions move through the electrolyte by migration and
diffusion from the positive electrode to the negative electrode (anode). At the
anode, the ions intercalate into the active particles. This process results in an
increase of the volume of the active particles, which in turn leads to a decrease in
porosity as well as flow of electrolyte out of the anode. Heat is generated during
the flow of current in the solid and liquid phases, flow of current in the
solid-liquid interface, and flow of ions in the electrolyte. During discharging, the
cycle is reversed.
Rechargeable lithium-ion batteries are widely used in a variety of applications, including
portable electronic devices and electric vehicles. The performance of a battery
highly depends on the effects of repeated charging and discharging cycles, which can
cause the degradation of the battery capacity over time. The porous electrode theory
( Newman et al., 2004) is commonly
accepted as the leading method for modeling the charge-discharge behavior of
lithium-ion cells. The method is based on a homogenized Newman-type approach that
does not consider the details of the pore geometry. The porous electrode theory is
based on a concurrent solution of a highly coupled multiphysics-multiscale
formulation. For more information, see Coupled Thermal-Electrochemical Analysis.
In some applications, a detailed understanding of the effects of the thermal-electrochemical
fields on the mechanical state (deformations) of the lithium-ion cell, electrolyte
fluid pressures, and electrolyte movement can be important to understand their
influence on the overall performance of the cell. Battery performance
characteristics (such as energy storage capacity and discharge voltages) can degrade
with deformations caused by particle swelling, thermal strains, and ion depletion.
Large deformations can cause failure of the separator, resulting in thermal runaways
in batteries. In such applications, the coupling between temperature and
displacement fields might be due to temperature-dependent material properties,
internal heat generation, or thermal expansion. The volume changes in the active
electrode particles resulting from the intercalation/deintercalation of lithium ions
are modeled as particle concentration–dependent eigenstrains at the macroscale
level. These volume changes affect the displacement field in the cell and porosity
and tortuosity evolutions. In addition, they generate a convective transport term in
the electrolyte that influences the movement of lithium ions through the
electrolyte.
The effects of fluid flow and fluid pressure are important to determine localized regions of
low electrolyte concentrations within the battery that can degrade battery
performance. It is possible to model the different regions in the battery as either
fully saturated or partially saturated to get a better understanding of electrolyte
flow effects during a charge-discharge cycle.
Governing Equations
The governing equations for the thermal-electrochemical process are based on the porous
electrode theory and are described in detail in Governing Equations. The governing equations for particle swelling are also described in Particle Swelling. The volumetric strain at the microscale, caused by particle swelling, results in
macroscopic eigenstrains (also referred to as inherent strain, assumed strain, or
"stress-free" strain) that affects the deformation of the porous electrodes, solid
volume fraction, , pore fluid pressure in the electrolyte, and electrolyte flow. Abaqus treats the continuum (porous electrodes containing electrolyte) as a multiphase
material and adopts an effective stress principle to describe its behavior (see
Effective stress principle for porous media). In this
approach, the total stress at a material point is assumed to be made up of an
effective stress, , which represents the stress in the skeleton of the porous
electrode, and an average pressure stress, , in the electrolyte, such that:
The above expression assumes partially saturated electrolyte flow, with saturation level, , where for fully saturated flow and represents partially saturated flow. The response of such a porous
medium is based on satisfying:
mechanical equilibrium based on all internal and external forces in the
system, as well as
fluid continuity for the electrolyte phase.
The porous medium is modeled with a Lagrangian finite element mesh that keeps track of the
evolving volume fractions of the solid and electrolyte phases at each material
point, as well as the flow of the electrolyte phase. The electrolyte is assumed to
be relatively incompressible. The effective response of the porous medium can be
described in terms of any mechanical constitutive model in Abaqus. For the special case of a linear elastic effective response, the effective
stress induced by an eigenstrain can be represented as:
where
is the effective Cauchy stress;
is the elasticity matrix;
is the total strain;
is the eigenstrain; and
is the elastic strain.
The continuity equation for the electrolyte flow is described in Continuity statement for the wetting liquid phase in a porous medium. The
formulation of the continuity equation is in terms of pore pressure as the basic
nodal variable (degree of freedom 8 at each node). The conjugate flux is the nodal
volumetric flow rate.
The ion flux equation (see Diffusion and Migration of Lithium Ions in the Liquid Phase) can optionally include advection contributions resulting from the velocity field
of the electrolyte. If advection effects are included in the model, any electrolyte
flux at the external boundaries is associated with a corresponding ion flux (that
is, the electrolyte leaves or enters the domain with ions). If advection effects are
not included in the model, only "clean electrolyte" (that is, electrolyte without
any ions) can leave or enter through the external boundaries.
Fully Coupled Solution Scheme
A fully coupled solution scheme is required when the stress analysis is strongly dependent on
the other fields involved in an electrochemical analysis, such as temperature, pore
pressure, electric potentials in the solid and electrolyte, and ion concentration.
In Abaqus/Standard, the temperature is integrated in time using a backward-difference scheme. The
nonlinear coupled system is solved using Newton's method. The coupled
thermal-electrochemical-structural–pore pressure analysis in Abaqus uses an exact implementation of Newton’s method, leading to an unsymmetric
Jacobian matrix in the form:
Steady-State Analysis
Steady-state analysis provides the steady-state solution by neglecting the transient
terms in the continuum scale equations. It can be used to achieve a balanced initial
state or to assess conditions in the cell after a long storage period.
A steady-state analysis neglects the transient terms in all the macroscopic field equations.
These include:
internal energy term in heat transfer;
transient term in the diffusion equation for the lithium ion concentration
in the electrolyte;
electrical transients in the conduction equations for both the solid and electrolyte phases
(characteristic time scales associated with the electrical transient effects
are much smaller compared to the characteristic times of thermal and
ion-diffusion); and
electrolyte flow continuity equation. When the modeling of electrolyte flow
includes gravity effects, a steady-state step ensures the loads and initial
stresses equilibrate exactly.
A steady-state analysis has no effect on the microscale solution; the transient terms
are always included in the solution of the lithium concentration in the solid
particle.
Input File Usage
Use the following option to define a steady-state fully coupled
thermal-electrochemical-structural–pore pressure analysis that considers the
fluid velocity effects:
Use the following option to define a steady-state fully coupled
thermal-electrochemical-structural–pore pressure analysis that ignores the fluid
velocity effects:
In a transient analysis, the transient effects in the heat transfer, pore fluid continuity,
and ion-diffusion equations are included in the solution. Electrical transient
effects are always omitted because they are rapid compared to the characteristic
times of thermal and ion-diffusion effects.
Input File Usage
Use the following option to define a transient fully coupled
thermal-electrochemical-structural–pore pressure analysis that considers the
fluid velocity effects:
Use the following option to define a transient fully coupled
thermal-electrochemical-structural–pore pressure analysis that ignores the fluid
velocity effects:
Spurious Oscillations Due to Small Time Increments
The integration procedure Abaqus/Standard uses for fluid flow analysis with the pore pressure degree of freedom introduces
a relationship between the minimum usable time increment and the element size. If
time increments smaller than these values are used, spurious oscillations can appear
in the solution.
The integration procedure Abaqus/Standard uses for the transient heat transfer equation also results in similar limitations
on the minimum usable time increment. By default, Abaqus/Standard uses nodal integration for the heat capacity term for first-order elements in a
transient analysis, resulting in a lumped treatment for this term. This treatment
eliminates nonphysical oscillations; however, it can lead to locally inaccurate
solutions, especially in terms of the heat flux for small time increments. If
smaller time increments are required, you should use a finer mesh in regions where
the temperature changes occur.
By default, the initial values of pore pressure, electric potential in the solid and
electrolyte, temperature, and ion concentration of all nodes are set to zero. You
can specify nonzero initial values for the primary solution variables (see Initial Conditions).
Boundary Conditions
You can prescribe the following boundary conditions:
Displacement (degrees of freedom 1, 2, and 3)
Pore pressure, (degree of freedom 8)
Electric potential in the solid, (degree of freedom 9)
Electric potential in the electrolyte, (degree of freedom 32)
Temperature, (degree of freedom 11)
Ion concentration in the electrolyte, (degree of freedom 33) at the nodes
You can specify boundary conditions as functions of time by referring to amplitude curves (see
Amplitude Curves).
A boundary without any prescribed boundary conditions corresponds to an insulated
(zero flux) surface.
The typical boundary condition consists of only grounding (setting to zero) the solid
electric potential at the anode. Thermal boundary conditions vary.
Loads
You can apply mechanical, pore fluid pressure, thermal, electrical, and electrochemical loads
in a coupled thermal-electrochemical-structural–pore pressure analysis.
Concentrated nodal forces on displacement degrees of freedom.
Distributed forces.
You can prescribe the following types of pore pressure loads (as described in Pore Fluid Flow):
Distributed pressure forces or body forces. The distributed load types available with
particular elements are described in the Abaqus Elements Guide. The magnitude and direction of gravitational loading
are usually defined by using the gravity distributed load type.
Pore fluid flow.
You can prescribe the following types of thermal loads (as described in Thermal Loads):
Concentrated heat flux.
Body flux and distributed surface flux.
Convective film and radiation conditions.
You can prescribe the following types of electrical loads on the solid (as described
in Electromagnetic Loads):
Concentrated current.
Distributed surface current densities and body current densities.
You can prescribe the following types of electrical loads on the electrolyte (as
described in Electromagnetic Loads):
Concentrated current.
Distributed surface current densities and body current densities.
You can prescribe the following types of ion concentration loads (as described in
Thermal Loads):
Concentrated flux.
Distributed body flux.
The typical loads include specification of an electric flux (current) in the solid phase of
the cathode. Thermal boundary conditions vary but typically include convective film
on the exterior surfaces. Typically, no loads are applied on the concentrations and
the electrolyte potential fields.
Predefined Fields
Predefined temperature fields are not allowed in coupled
thermal-electrochemical-structural–pore pressure analyses. Instead, you can use
boundary conditions to prescribe degree of freedom 11 for temperature. You can
specify other predefined field variables in a fully coupled
thermal-electrochemical-structural–pore pressure analysis. These values affect only
field variable–dependent material properties, if any.
Material Options
In a coupled thermal-electrochemical-structural-pore pressure analysis, you can specify the
effective response of the medium in terms of any available mechanical constitutive
behavior. The thermal and electrical properties for both the solid and the
electrolyte phases are also active during this analysis.
The electrochemistry framework requires that the material definition contain the complete
specification of properties required for the porous electrode theory, as described
in Coupled Thermal-Electrochemical Analysis. In addition, the material name
must begin with "ABQ_EChemPET_" to enable the coupled micro-macro solution at the
different electrodes. Special-purpose parameter and property tables of type names
starting with “ABQ_EChemPET_” are required in these material definitions (see Parameter Table Type Reference and Property Table Type Reference). For more information about the material definitions for the
thermal-electrochemical behavior, see Material Options.
In problems formulated in terms of total pore pressure, you must include the density of the
dry material in the material definition (see Density). You must specify the permeability of the porous medium
including, optionally, its dependence on the void ratio, saturation, and the flow
velocity (see Permeability). You must also define the specific weight of the electrolyte as
part of the material permeability definition. You can define the compressibility of
the solid grains and of the electrolyte for both fully and partially saturated flow
problems (see Elastic Behavior of Porous Materials). If you
do not specify the compressibility of the solid and the liquid phases, they are
assumed to be fully incompressible.
For partially saturated flow, you must define the porous medium's absorption/exsorption
behavior (see Sorption).
Elements
A fully coupled thermal-electrochemical-structural–pore pressure analysis requires the use of
elements that have displacement (degrees of freedom 1, 2, and 3), pore pressure
(degree of freedom 8), electric potential in the solid (degree of freedom 9),
temperature (degree of freedom 11), electric potential in the electrolyte (degree of
freedom 32), and ion concentration in the electrolyte (degree of freedom 33) as
nodal variables. The coupled thermal-electrochemical-structural–pore pressure
elements are available in Abaqus/Standard only in three dimensions (see Coupled Thermal-Electrochemical-Structural–Pore Pressure Elements).
Output
In addition to the output variables available for the coupled thermal-electric, the coupled
thermal-electrochemical, and coupled thermal-electrochemical structural procedures,
you can request the Abaqus/Standard output variables related to pore pressure degrees of freedom (see Output).